17. If 9 persons use 13 pounds of tea in a month, how much will 10 persons use in a year? 18. If of of a gallon of wine cost of a dollar, what will 5 gallons cost? 19. How many yards of carpeting, 1 yards wide, will it take to cover a floor that is 4 yards wide and 6 and threefifths yards long? 20. Three persons bought a hogshead of sugar containing 413 pounds. The first paid $21 as often as the second paid $33, and as often as the third paid $4: what was each one's share of the sugar? 21. A, with the assistance of B, can build a wall 2 feet wide, 3 feet high, and 30 feet long, in 4 days; but with the assistance of C, they can do it in 3 days: in how many days can C do it alone? 22. If two persons engage in a business, where one advances $875, and the other $625, and they gain $300, what is each one's share. 23. A person purchased of a vessel, and divided it into 5 equal shares, and sold each of those shares for $1200: what was the value of the whole vessel? 24. How many yards of paper, of a yard wide, will be sufficient to paper a room 10 yards square and 3 yards high? 25. What will be the cost of 45lbs. of coffee, New Jersey currency, if 9lbs. cost 27 shillings? 26. What will be the cost of 3 barrels of sugar, ing 2cwt. at 10d. per pound, Illinois currency? each weigh 27. If 12 men reap 80 acres in 6 days, in how many days will 25 men reap 200 acres? 28. If 4 men are paid 24 dollars for 3 days' labor, how many men may be employed 16 days for $96? 29. If $25 will supply a family with flour at $7.50 a barrel for 2 months, how long would $45 last the same family when flour is worth $6.75 per barrel? 30. A wall to be built to the height of 27 feet, was raised to the height of 9 feet by 12 men in 6 days: how many men must be employed to finish the wall in 4 days at the same rate of working? 31. A, B and C, sent a drove of hogs to market, of which A owned 105, B 75, and C 120. On the way 60 died: how many must each lose? 32. Three men, A, B and C, agree to do a piece of work, for which they are to receive $315. A works 8 days, 101 hours a day; B 92 days, 8 hours a day; and C, 4 days, 12 hours a day what is each one's share? : 33. If 10 barrels of apples will pay for 5 cords of wood, and 12 cords of wood for 4 tons of hay, how many barrels of apples will pay for 9 tons of hay? 34. Out of a cistern that is full is drawn 140 gallons, when it is found to be full: how much does it hold? 35. If .7 of a gallon of wine cost $2.25, what will .25 of a gallon cost? 36. If it take 5.1 yards of cloth, 1.25 yards wide, to make a gentleman's cloak, how much surge, yards wide, will be required to line it? 37. A and B have the same income. A saves of his annually; but B, by spending $200 a year more than A, at the end of 5 years find himself $160 in debt: what is their income? 38. A father gave his younger son $420, which was of what he gave to his elder son; and 3 times the elder son's portion was the value of the father's estate what was the value of the estate? 39. Divide $176.40 among 3 persons, so that the first shall have twice as much as the second, and the third three times as much as the first: what is each one's share? 40. A gentleman having a purse of money, gave of it for a span of horses; of of the remainder for a carriage: when he found that he had but $100 left: how much was in his purse before any was taken out? 41. A merchant tailor bought a number of pieces of cloth, each containing 25 yards, at the rate of 3 yards for 4 dollars, and sold them at the rate of 5 yards for 13 dollars, and gained by the operation 96 dollars: how many pieces did he buy? RATIO AND PROPORTION. 221. Two numbers having the same unit, may be com pared in two ways: 1st. By considering how much one is greater or less than the other, which is shown by their difference; and, 2d. By considering how many times one is contained in the other, which is shown by their quotient. In comparing two numbers, one with the other, by means of their difference, the less is always taken from the greater. In comparing two numbers, one with the other, by means of their quotient, one of them must be regarded as a standard which measures the other, and the quotient which arises by dividing by the standard, is called the ratio. 222. Every ratio is derived from two numbers: the first is called the antecedent, and the second the consequent: each is called a term, and the two, taken together, are called a couplet. The antecedent will be regarded as the standard. If the numbers 3 and 12 be compared by their difference, the result of the comparison will be 9; for, 12 exceeds 3 by 9. If they are compared by means of their quotient, the result will be 4; for, 3 is contained in 12, 4 times: that is, 3 measuring 12, gives 4. 223. The ratio of one number to another is expressed in two ways: 1st. By a colon; thus, 3 : 12; and is read, 3 is to 12; or, 3 measuring 12. 2d. In a fractional form, as 12 or, 3 measuring 12. 221. In how many ways may two numbers, having the same unit, be compared with each other? If you compare by their difference, how do you find it? If you compare by the quotient, how do you regard one of the numbers? What is the ratio? 222. From how many terms is a ratio derived? What is the first term called? What is the second called? Which is the standard? 293. How may the ratio of two numbers be expressed? How read? 224. If two couplets have the same ratio, their terms are said to be proportional: the couplets 3 : 12 and 1 : 4 have the same ratio 4; hence, the terms are proportional, and are written, 3 : 12 : : 1 : 4 by simply placing a double colon between the couplets. The terms are read 3 is to 12 as 1 is to 4, and taken together, they are called a proportion: hence, A proportion is a comparison of the terms of two equal ratios.* 224. If two couplets have the same Ilow are they written? How read? ratio, what is said of the terms? What is a proportion? *Some authors, of high authority, make the consequent the standard and divide the antecedent by it to determine the ratio of the couplet. The ratio 3: 12 is the same as that of 1: 4 by both methods; for, if the antecedent be made the standard, the ratio is 4; if the consequent be made the standard, the ratio is one-fourth. The question is, which method should be adopted? The unit 1 is the number from which all other numbers are derived, and by which they are measured. The question is, how do we most readily apprehend and express the relation between 1 and 4? Ask a child, and he will answer, "the difference is 3." But when you ask him, "how many 1's are there in 4?" he will answer, "4," using 1 as the standard. Thus, we begin to teach by using the standard 1: that is, by dividing 4 by 1. Now, the relation between 3 and 12 is the same as that between 1 and 4; if then, we divide 4 by 1, we must also divide 12 by 3. Do we, indeed, clearly apprehend the ratio of 3 to 12, until we have referred to 1 as a standard Is the mind satisfied until it has clearly perceived that the ratio of 3 to 12 is the same as that of 1 to 4? In the Rule of Three we always look for the result in the 4th term. Now, if we wish to find the ratio of 3 to 12, by referring to 1 as & £tandard, we have 3 : 12 : : 1 : ratio, which brings the result in the right place. But if we define ratio to be the antecedent divided by the consequent, we should have 3 : 12 : : ratio : 1, which would bring the ratio, or required number, in the 3d place. 225. The 1st and 4th terms of a proportion are called the extremes: the 2d and 3d terms, the means. Thus, in the portion, 3 : 12 : : 6 : 24 3 and 24 are the extremes, and 12 and 6 the means: pro we shall have, by reducing to a common denominator, But since the fractions are equal, and have the same denominators, their numerators must be equal, viz. ; In any proportion, the product of the extremes is equal to the product of the means. 226. Since, in any proportion, the product of the extremes is equal to the product of the means, it follows that, In all cases, the numerical value of a quantity is the number of times which that quantity contains an assured standard, called its unit of measure If we would find that numerical value, in its right place, we must say, standard : quantity : : 1 : numerical value: but if we take the other method, we have quantity : standard : : numerical value : 1, which brings the numerical value in the wrong place. |